ESTIMATION OK l< V 1 > V It RANGES OVER THE OPEN OCEAN 



257 



& = 



2d 

 L 



L is an abbreviation for (aX 2 /67r 2 )* and is equal to 

 33 ft for X = 10 cm, 



Tm is the characteristic value for the standard case, 

 7m is the characteristic value for the parabolic case, 

 '2\* 



h 2 {z) is 



j) **«(!*). 



//j ( ' 2) is the Hankel function of second kind, order 

 }4, of the argument ( ^2= 



e m 's are roots of /j 2 (f) = 0. 



The expression above has been evaluated for the 

 first mode by summing the series for /i 2 and perform- 

 ing the integration numerically. This curve is remark- 

 able for the considerable interval in which the 

 ordinate is practically zero. The attenuation constant 

 differs by less than 1 per cent from the standard for 

 ducts below 5 = 1.2. Beyond this value the effect of 

 the duct increases rapidly, and when 5 = 1.7 the 

 attenuation constant is 10 per cent different from 

 standard, and at 5 = 2 it is 20 per cent different. 



It seems that at least for radar purposes the 

 condition 5 < 1 is a reasonable and convenient 

 condition for defining a negligible duct. This is 

 equivalent to saying that L/2 is the thickness below 

 which a duct may for practical purposes be disre- 

 garded. For instance, at X = 10 cm, L = 33 ft, and 

 hence we conclude that the effect of ducts less than 

 16 ft in thickness on 10-cm radars may be neglected. 

 On the other hand, if the wavelength is 3 m, L = 300 

 ft, and ducts below 150 ft in thickness are negligible. 



If in the interest of simplicity we neglect the effect 

 of small variations in the characteristic values on 

 the characteristic functions, the fractional change in 

 attenuation constant is also equal to the fractional 

 change in the range against surface targets. It follows 

 that the estimation of range can be reduced to a 

 measurement of sea temperature and specific hu- 

 midity at masthead level; for the duct thickness d 

 under conditions of neutral equilibrium is given by 



(q* - q a ) T 



d = - 



q s is the saturation specific humidity at sea tempera- 

 ture and q a the specific humidity at masthead. Y is 

 a parameter for which a representative value is 

 0.08, and (dq/dz) Q is the gradient of specific humidity 

 required to give zero M gradient under conditions 



of constant potential temperature. It is taken as 

 Yi g per kg. 

 Thus it turns out that 



6 = 0.32 q " ~ CU 



Li 



where - J — - — - is in grams per kg per 100 ft . 



If L is given the appropriate value for X = 10 cm 



5 ^ q, - q a (g per kg) . 



For illustrative purposes, scales of (q s — q a )/L and 

 q s — q a for X = 10 cm have been added in Figure 2. 



Figure 2. Fractional drop in attenuation constant of 

 the first mode versus duct thickness. Bottom scale for 

 A q and A q/L corresponds to X = 10 cm. 



It is emphasized that the calculations are rough 

 and are presented only in the belief that some sort 

 of simple guiding principle may be more useful than 

 a highly accurate and cumbersome formula. The 

 results given are accurate out to variations in range 

 of 1 per cent, and the determination of threshold 

 thickness is completely reliable. Extension beyond 

 5 = 1.2 is a definite extrapolation. The trend indi- 

 cating that the increase in range goes up at least as 

 fast as the sixth power of the duct thickness for 

 5 > 1 is, we believe, real. 



